7th Grade - Chapter 11 : Ratios, Percent and Proportions
Objectives:
Estimate Percent
Express ratios and rates as fractions
Solve proportions by using “Cross Products”
Solve problems by drawing diagrams
Express numbers as percent, fractions or decimal
Find a percent of a number
PERCENT / DECIMAL / FRACTION TABLE:
𝟏
𝟐
10%
10
1
=
= .𝟏
100 10
12 𝟐%
12.5
1
= =. 𝟏𝟐𝟓
100
8
25%
25
1
= =. 𝟐𝟓
100
4
33 𝟑%
33.333
1
= =. 𝟑𝟑𝟑
100
3
60%
60
3
= =. 𝟔
100
5
37 𝟐%
37.5
3
= =. 𝟑𝟕𝟓
100
8
80%
80
4
= =. 𝟖
100
5
87𝟐%
87.5
7
= =. 𝟖𝟕𝟓
100
8
50%
50
1
= =. 𝟓
100
2
75%
75
3
= =. 𝟕𝟓
100
4
𝟏
16 𝟑%
16.66
1
= = .166
100
6
𝟏
𝟏
62 𝟐%
62.5
5
= =. 𝟔𝟐𝟓
100
8
𝟏
20%
20
1
= =. 𝟐
100
5
40%
40
2
= =. 𝟒
100
5
𝟐
66 𝟑%
66.66
2
= =. 𝟔𝟕
100
3
Remember:
.𝟓
.5% = 𝟏𝟎𝟎 = .005
Decimal % is small!
Ratios, proportions, percent and rate are all ways of expressing and comparing numbers.
60
60% means:
: Percent means PER 100 (century, cents, centimeters).
100
Ratios are when you compare 1 thing to another by division (setting up a fraction.)
𝑃𝑎𝑟𝑡
𝑑𝑜𝑔𝑠
𝑃𝑎𝑟𝑡
𝑑𝑜𝑔𝑠
Usually a
𝑒𝑥𝑎𝑚𝑝𝑙𝑒:
𝑜𝑟
example:
.
𝑊ℎ𝑜𝑙𝑒
𝐴𝑙𝑙 𝑝𝑒𝑡𝑠
It can also used to compare
𝑃𝑎𝑟𝑡
𝑵𝒆𝒘
𝑶𝒓𝒊𝒈𝒊𝒏𝒂𝒍
𝑐𝑎𝑡𝑠
for example: Sales Price to Original.
Another way to think of a Ratio is also a “probability”? Or what are the “Good
Results” vs the “Total Number” of results. Roll of Dice, picking a card from a deck,
chance of winning a raffle:
Ratios can also be used to compare two items:
This is called a RATE:
𝑰𝒕𝒆𝒎𝟏
Something
Something
or
𝑰𝒕𝒆𝒎𝟐
You did this when you compared fractions. You must make sure your UNITS are the same:
(for example: Dollars or Cents or Ounces or Pounds, Minutes or Hours, Inches and Feet, etc.)
per
You also use Ratios (RATES) when you compare prices of items in the grocery store (or express one item as
a PER another item). Miles per Hour, Cost per Pound.
7th Grade - Chapter 11 : Ratios, Percent and Proportions
ESTIMATING: Many times you can “Estimate” for the calculation you are trying to solve.
1
Sometimes the Fraction is Easier: Example 33% is about .
3
ESTIMATING IT IS NOT THE SAME AS ROUNDING!
You don’t always have to get an accurate value!
The word “ABOUT” is also a key that you don’t need an actual answer!
There is an ART to Estimating. Pick numbers that are easy to solve and come close!
A) 22% of 197 is about how much?
20
1
22% is close to 20%
20% = 100 = 5 =. 𝟐
197 is close to 200
1
200
So: .2 x 200 = 40
or 5 𝑥 200 = 5 = 40
B) 10% technique: 10% of 200 = 20 so (20% is Twice 10%) or 2 x 20 = 40
1
𝟏.𝟓𝟎𝟎
C) 50% of 1,512
50% = 2 =. 𝟓
1,512 is about 1,500, So: 𝟐 = 𝟕𝟓𝟎
PERCENT OF A NUMBER:
There are many situations in life where you need to know various relationships about
numbers. Very often, the relationship is expressed in terms of a percent.
A machine produced 2,500 units per day. Production is going to be reduced by 20%. How
many units will be made?
20
1
𝟏
Method 1: 20% = 100 = 5. So 𝟓 of 2,500 = 500 Units less. Therefore the output will be 2,000 Units.
Method 2: 20% = .2 so: .2 x 2,500 = 500 Units less. The output will be 2,500 – 500 = 2,000 Units.
Method 3: Cross Multiply Method:
𝟐𝟎
𝟏𝟎𝟎
=
𝒓
𝟐,𝟓𝟎𝟎
𝑤ℎ𝑒𝑟𝑒 𝑟 = 𝑢𝑛𝑖𝑡𝑠 𝑡𝑜 𝑟𝑒𝑑𝑢𝑐𝑒: 100r = 50,000. So r = 500
Method 4: Cross Multiply Method: since you are going to reduce output by 20%. The NEW output
will be 80% of the original so:
80
=
100
𝑁
2,500
𝑤ℎ𝑒𝑟𝑒 𝑁 = 𝑖𝑠 𝑡ℎ𝑒 𝑁𝐸𝑊 𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑜𝑢𝑡𝑝𝑢𝑡.
100N = 200,000 so N = 2,000 units.
Method 5: 80% = .8 so .8 x 2,500 = 2,000 new units
RATIOS can always be set up:
𝑷𝒂𝒓𝒕
=
𝑩𝒂𝒔𝒆
?%
𝟏𝟎𝟎
BASE could either be a “TOTAL” or “Original” value
PART could be the CHANGE AMOUNT or the NEW AMOUNT
The
?%
𝟏𝟎𝟎
𝒊𝒔 𝒔𝒐𝒎𝒆𝒕𝒊𝒎𝒆𝒔 𝒄𝒂𝒍𝒍𝒆𝒅 𝒕𝒉𝒆 𝑹𝒂𝒕𝒆
7th Grade - Chapter 11 : Ratios, Percent and Proportions
PROPORTIONS: An equation that shows if two ratios are equivalent. (Denominators≠ 0)
Proportions were used when you solved for common denominators. You compared one fraction to
another (You made them equal). But sometimes you need to find out if two items are equal.
The Common Denominator method works when one of the numbers is a multiple of the other but
you need a way to ALWAYS find if proportions are equal.
Cross Products: The “Cross Products” of a proportion are equal.
IF
𝒂
𝒃
=
𝒄
𝑻𝑯𝑬𝑵 𝒂𝒅 = 𝒃𝒄
𝒅
The Shadow Problem: John is 5 feet tall and his shadow is 15 feet long. A Flag Pole has
a shadow 75 feet long. How tall (t) is the Flag Pole?
RATIO:
𝐻𝑒𝑖𝑔ℎ𝑡
𝑆𝑜:
𝑆ℎ𝑎𝑑𝑜𝑤
𝐽𝑜ℎ𝑛 5
15
=
𝑃𝑜𝑙𝑒 (𝑡)
𝑠𝑜: 15𝑡 = 5 𝑥 75 𝑜𝑟 15𝑡 = 375 ∶
75
375
15
= 25 𝑓𝑒𝑒𝑡
Percent, Decimals and Fractions:
Sometimes it is easier to work with the Decimal Value, the Fraction or the Percent.
28
14
7
28% means 28 out of 100 or
=
=
100
50
28
25
𝟏
Fractions can be written as decimals:
= . 28
= . 𝟐 = 𝟐𝟎%
100
𝟓
Common Denominator: Equivalent Fractions - Only works if one value is a multiple!
1
5
𝑥
?
?
=
𝑥
So:
100
Cross Product: Always works:
1
5
=
𝑋
100
1
5
𝑥
20
20
20
=
100
= 20%
But sometimes the math is harder.
So, 5x = 1 x 100; or 5x = 100. Divide both sides by 5: x =
100
5
= 20%
𝟏𝟕
Example: If you score a 17 out of 20 on a test. The ratio is: 𝟐𝟎.
If you wanted to know what percent that is, you could:
1) Set up Common Denominator: Equivalent Fractions.
2) Using the Cross Multiplication Technique:
So, 20x = 1,700;
17
20
=
𝑋
100
17
20
𝑥
5
=
5
85
100
= 85%
:
Divide both sides by 20: x =
1,700
3) Calculate the decimal: 20|17 = .85 𝑎𝑠 𝑎 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 =
20
85
=
100
170
2
= 85%
𝑜𝑟 85%
Be careful with a Percent that has a decimal: 5% and .5% are very different!
5% =
𝟓
𝟏𝟎𝟎
= .05
.5% =
.𝟓
𝟏𝟎𝟎
= .005
7th Grade - Chapter 11 : Ratios, Percent and Proportions
PIE Chart Percentages: Pie Charts are useful graphical tools to showing and
representing data. Once you have collected the data and calculated the fractional
or decimal value for each part, you set up an equivalent fraction to make a ratio
using 360 as the base since a circle has 360O. This example has 4 values: A, B, and C
and D. (numbers are made up for easy math) But in reality, the TOTAL could be any
value and the Values won’t always be nice numbers.
1) Calculate the Total: 25 + 15 + 5 + 55 = 100 (or whatever the data is!)
𝐼𝑡𝑒𝑚 𝑣𝑎𝑙𝑢𝑒
2) Create a Ratio or Fraction for each Part (ITEM) :
𝑇𝑜𝑡𝑎𝑙
3) Sometimes it is easier to do the calculation as a decimal rather than a fraction.
(Fractions will get you a better result if the denominator is a strange” total.)
4) Since a Circle (pie) has 360 degrees, each portion of the pie has an equivalent
number of degrees.
Degree Equivalent
Item
Value Fraction
Decimal
When DRAWING, you can round if needed
A
25
25/100
.25
B
15
15/100
.15
C
5
5 /100
.05
D
55
55/100
.55
TOTAL
100
25
𝑥 360 𝑜𝑟 .25 𝑥 360 = 90
100
15
𝑥 360 𝑜𝑟 .15 𝑥 360 = 54
100
5
𝑥 360 𝑜𝑟 .05 𝑥 360 = 18
100
55
𝑥 360 𝑜𝑟 .55 𝑥 360 = 198
100
7th Grade - Chapter 11 : Ratios, Percent and Proportions
USING STATISTICS TO PREDICT:
5
5
In the M&M example, many students got: 19 𝑟𝑒𝑑 𝑀&𝑀𝑠 in their bags. This is approximately 25% since 19
5
1
is about 20 = 4. If we were to open and pour many bags of M&Ms into a bowl and pull them out
randomly, we would expect to get 25% to be red.
You often cannot measure the entire Population. How would you count ALL the fish in a lake? You cannot
ask every person in a city if they like a certain candy bar. Population is the TOTAL amount available
combined.
The Sample is a way to select items from the population that you think is representative of the population
in Total. There is an entire area of math called STATISTICS that deals with how to select a valid sample.
You need to be careful of BIAS when selecting your sample.
Random means that each person from the population has an equal chance of being picked. If you were to
stand in front of Wal-mart and ask a question, you might get a very different response from people going
into Target.
A Pizza company offered a new flavor of pizza to students at a middle school and 8 or of 10 students
liked it. The company thought it would be a great idea to offer this on their menu in the city. Is this a
good idea?
The same Pizza Company wanted to take their pizza to another middle school with 900 students.
8
𝑥
About how many students at the new middle school might like the pizza? 10 = 900 𝑜𝑟 .8 𝑥 900.
In a survey, 5 out of 12 dentists think Brand-X is a good toothpaste. There are 2,400 dentists in the
5
𝑥
state. About how many might be in favor of toothpaste? =
𝑜𝑟 12𝑥 = 12,000: 𝑥 = 1,000.
12
2,400
In a recent election, when leaving the voting polls, 2 out of 5 people said they voted for a certain
candidate. There are 700 people who could vote. How many votes might the candidate be expected to
2
𝑥
2
get? ? 5 = 700 𝑜𝑟 5𝑥 = 1,400: 𝑥 = 280 OR: 5 = .4 𝑠𝑜 .4 𝑥 700 = 280
In a sample of fish taken from a lake, 30% were female. In another sample of 600 fish, how many
might be expected to be male? 30% = .3 so .3 x 600 = 180 might be males.
𝑥
30
18,000
Or 600 = 100 so: 100x = 30 x 600 : 100x = 18,000 : 100 = 180
7th Grade - Chapter 11 : Ratios, Percent and Proportions
Percent Change
Percent of Change compares an Original Value (Base) to some number that
has been Increased or Decreased.
The Greek Letter: Δ (DELTA) is often used to mean change in:
Example #1) A dress was 40 dollars and it is on sale for $36. What is the percent discount?
The Original value (Base) is $40, The Δ (DELTA) Change is 40 - 36 or $4)
𝑠𝑜:
Δ
𝑏𝑎𝑠𝑒
=
𝑥
100
or:
40x = 400 or x =
4
40
400
40
𝑥
=
100
= 10%
Example #2) School has 300 Students. They plan to add a grade next year and have 360 students. What
is the percent increase for next year?
The Original value (Base) is 300, The Δ (DELTA) Change is: 360 - 300 or 60 Students.
𝑠𝑜:
Δ
𝑏𝑎𝑠𝑒
=
𝑥
100
or:
60
=
300
5x = 100 or x =
100
5
𝑥
100
1
𝑥
5
100
or =
= 20%
Example #3) John was 4’ 6’ at the start of the year. He grew 4 inches. What percent did he grow? (The
“trick” to this problem is to make sure your units are the same: Either Inches or feet): 4’, 6” or 54 Inches.
The Original value (Base) is 54, The Δ (DELTA) Change is 4)
𝑠𝑜:
Δ
𝑏𝑎𝑠𝑒
=
𝑥
100
or:
4
54
=
27x = 200 or x =
𝑥
100
200
27
or
2
27
=
𝑥
100
= 7.4%
Estimate: 10% growth would be 5.4 inches